from the qubit to the quantum search algorithms

FROM THE QUBIT TO THEQUANTUM SEARCH ALGORITHMS

Prof. Gianfranco CariolaroTommaso Occhipinti

Erice, 18/04/2007

Tommaso Occhipinti, DAA Erice2

Outline

• Quantum Mechanics• Quantum states

– Qubit– Coherent states

• Qubit properties

• Quantum Algorithms• Quantum Applications

– Computers– Cryptography– Communications– Astronomy

Part 1

Part 2


Free SpaceQuantum Cryptography

The Quantum Optics Instrument for OWL

QuantumAstronomy Instruments

Introduction

QuantumMechanics applied to

Information


QM - Postulate 1

Postulate 1 gives us the universal mathematicalmodel of any physical system: a vector Hilbertspace on the complex numbers

[1] P.A.M. Dirac: “The Principles Of Quantum Mechanics”, Oxford University Press (1958).


QM - Postulate 2

describes the temporal evolution of a closedphysical system;


The state of the system afterthe measurement is:

QM - Postulate 3

is about “quantummeasurements” andindicates the way toextract informationfrom a quantumsystem in a preciseinstant


QM - Postulate 4

formalizes the interaction of manyphysical systems with the combination ofdifferent Hilbert spaces coming to aunique Hilbert space.


Quantum States for Information

The Quantum BitQUBIT

CoherentStates

Usually the qubit is made for:• Computation• Security Key distributions

the Coherent State ismade for:• QuantumCommunications


Qubit definition

We use the BRACKET notation(by P.A.M. Dirac)

is a “ket”

is a “bra”Is the inner product between theket vector and the ket

Notation example:


Some Qubit properties

Some Properties• Quantum measurement• Superposition• Entanglement• No cloning• Indistinguishibility of non-

orthogonal states

OneQubit

TwoQubits


Qubit in practice


Take the overall state of two qubits, and supposethat these qubits are in a superposition of the basis

states |0> and |1>. The state is:

We evaluated value of f(.) for all the possible basisfour states in only one stepQUANTUM PARALLELISM

Example: Two qubits calculus

f(.)Suppose to apply a logical gate f(.) to this state

Second PostulateFourth Postulate


There are ALL the results

But, measuring, we obtainthem with a particular

probability

… the trick is to find the goodcombination of quantum gatesthat can bring the state of thesystem to give the searched

result In this sense a quantumalgorithm is a:

classical PROBABILISTIC algorithmPLUS

the good properties and thelimitations of Quantum Mechanics

Quantum Algorithms


Quantum Algorithms examples

Shor’s Quantum Algorithm (1994)It makes the factorization of big integer numbers of dimension n.It is

extremely efficient O(n³)Classically the computational complexity is exponential

Grover’s Algorithm

Efficient way of finding an object inside an unsorted DataBase onN elements in O(sqrt(N))

QFT (Quantum Fourier Transform)


The Known Quantum Algorithm

[2] M.A. Nielsen, I.L. Chuang: “Quantum Computation and Quantum Information”,Cambridge: University Press, 2000.

Designing a new quantum algorithm is verydifficult and anti-intuitive


Some Quantum Applications

1. Quantum Computers (where quantumalgorithms runs)

2. Quantum Cryptography (the rules ofnature assure the secrecy of the transmissionof classical information)

3. Quantum Communication (Tx and Rxcapable of reaching the limit of communicationstheory)

4. Quantum Astronomy (finding the natureof the light coming from astronomical sources)


Quantum Computing

The qubits as the elementaryunits in computation…

…we can realize theQuantum Computer (QC)

Inside it we can:

• Write• Perform several operations

• Read

[2] M.A. Nielsen, I.L. Chuang: “Quantum Computation and Quantum Information”,Cambridge: University Press, 2000.[5] J. von Neumann, Mathematical Foundation of Quantum Mechanics, Princeton Univ. Press, Princeton 1955.

Postulate 3Postulate 2

With the postulates ofquantum mechanics wedescribe the functionalitiesof a QC

DWAVE copyright


Quantum Key Distribution

We think that QKD is really a telecommunicationsystem, a protocol that can be described like manyother classical network protocol

It is totallySecure

It has been demonstrated that QKD has a big property: theunconditional security. It is secure against any type of attack,

even the future ones, even if the attacker has infinitecomputational power and infinite quantity of money!


Quantum Communications

ClassicalSource

QuantumCoder

QuantumCommunication

Channel

QuantumMeasurement

Decisionon theSymbol

Quantum TX Quantum RX

Laser producingVery goodCoherent States

Glauber RJ (1963a), Photon Correlations, Phys. Rev. Letters vol. 10, 84

Quantum Communicationsare more efficientcompared to the classicaloptical communications

C.-W. Lau, et al., Binary Quantum Receiver Concept Demonstration, IPN Progress Report 42-165

Beam Splitter

Single photon detector


Quantum Astronomy

Results

Quantum Measurements

We measure the arrivaltimes of single photons atthe level of picoseconds.

Filtering in wavelength andpolarization.

Where are the Quantum States?

Quantum Statistics(CLASSICAL Search)

Search for correlations in arrivaltime and in different polarizationchannels.Moreover the same analysis can beaccomplished on the data acquiredby another twin telescope.

Quantum Astronomy Investigation

Acquisition Storage DataAnalysis


Quantum Astronomy: FUTURE???

Quantum Astronomy Investigation

Acquisition Storage DataAnalysis

Quantum Information

We would need:

Quantum Memories orHybrid Quantum memories

In order to input the classicaldata inside a quantum computer

Quantum Memory

Quantum Computer 1. Search with Grover’salgorithm looking for somepattern inside the streamof photons?

2. Compute some calculationstaking advantage from theQuantum FourierTransform?


Conclusions

a. With the quantum states it is possible tointroduce innovative techniques

a. Qubit for the elaboration of informationa. Quantum Computersb. Quantum Algorithms

b. Transmission of coherent states forcommunicationa. Ex. Quantum Phase Shift Keying (QuPSK)

b. Quantum Astronomy can be developedtaking into account both the qubit ideaand the techniques of quantumcommunications


Thanks

Tommaso [email protected]


Bibliography1. P.A.M. Dirac: “The Principles Of Quantum Mechanics”, Oxford University Press (1958).2. M.A. Nielsen, I.L. Chuang: “Quantum Computation and Quantum Information”,Cambridge: University Press, 2000.3. C. Shannon “Communication Theory of Secrecy Systems”, Bell System Technical Journal, vol.28(4), page 656-715, 1949.4. A. Einstein, B. Podolsky, N. Rosen: “Can quantum-mechanical description of physical reality be considered complete?” Physical

Review 41, 777 (15 May 1935).5. J. von Neumann, Mathematical Foundation of Quantum Mechanics, Princeton Univ. Press, Princeton 1955.6. J.S. Bell “On the Einstein Podolsky Rosen paradox” Physics 1 n.3, 195 (1964).7. W.K. Wootters, W.H. Zurek, “A Single Quantum cannot be Cloned”, Nature, Vol.299, pp.802 - 803, 28 October 1982.8. B. Schneier, Applied Cryptography: Protocols, Algorithms, and Source Code in C, Second Edition, Wiley and Sons, 19969. A. S. Tanenbaum, Computer Networks, Prantice Hall PTR, 199610. R. Rivest, A. Shamir, L. Adleman, A Method for Obtaining Digital Signatures and Public-Key Cryptosystems, Communications ACM,

vol.28, pp.120-134, 1978.11. R. J. Hughes, G. L. Morgan, C. G. Peterson, “Practical Quantum Key Distribution Over a 48-km Optical12. C. H. Bennet and G. Brassard and J. M. Robert, Privacy Amplification by Public Discussion, SIAM Journal on Computing, 2,17, 199813. C. H. Bennet and G. Brassard and C. Crépeau and U. M. Maurer, Generalized privacy amplification, IEEE Trans. Information Theory,

vol. 41, 1995, Fiber Network”, arXiv e-print archive: 9904038, 1999.14. G. Gilbert, M. Hamrick: “Practical Quantum Cryptography: A Comprehensive Analysis (Part One)”, Mitre Technical Report, settembre

2000.15. Nicolas Gisin, Gregoire Ribordy, Wolfgang Tittel, Hugo Zbinden: “Quantum Cryptography”, arXiv quant-ph 0101098 v2, 18 set

2001.16. D. Mayers: “Unconditional security in quantum cryptography”, Acm Journal, vol 48, pp. 351-406, 2001.17. H.Inamori, N. Lutkenhaus, D. Mayers: “Uncoditional Security Quantum Key Distribution”, arXiv e-print archive, 200118. R. Renner, N. Gisin, B. Kraus: “An Information-Theoric Security Proof for QKD Protocols”, arXiv e-print archive, 200519. Sergio Cova: “Single-Photon Photon Avalanche Diodes: Retrospect and Prospect”.20. S.Cova, A.Longoni and G.Ripamonti “Active-Quenching and Gating Circuits for Single-Photon Avalanche Diodes (SPADs)” IEEE

Transactions on Nuclear Science, NS-29, pp. 599-601 (1982).21. Glauber RJ (1963a), Photon Correlations, Phys. Rev. Letters vol. 10, 8422. Glauber RJ (1963b), The Quantum Theory of Optical Coherence, Phys. Rev. vol. 130, 252923. Glauber RJ (1963c), Coherent and incoherent states of the radiation field, Phys.Rev. 131, 2766-2788 (1963)24. Hanbury Brown R, (1974), The Intensity Interferometer. Taylor and Francis, New York.25. T. Occhipinti: “Comunicazioni Quantistiche”, Padova: Copisteria, Portello, 2005.26. C. Barbieri, G. Cariolaro, T. Occhipinti, C. Pernechele, F. Tamburini and P. Villoresi, Qspace Project: Quantum Cryptography in

Space Optical Communication theory and techniques ed. Enrico Forestieri, Springer 200427. Naletto G, Barbieri C, Dravins D, Occhipinti T, Tamburini F, Da Deppo, V, Fornasier S, D’Onofrio M, Fosbury RAE, Nilsson R, Uthas H,

Zampieri L, (2006), QuantEYE: A Quantum Optics Instrument for Extremely Large Telescopes, In: Ground-Based and AirborneInstrumentation For Astronomy, SPIE Proc. Vol. 6269.

28. D. L. Perry “VHDL, Programming by example”, quarta edizione, McGraw-Hill 2002.29. Xilinx, PowerPC 405 Processor Block Reference Guide Embedded Development Kit UG018 (v2.1) July 20, 2005.30. Xilinx, Virtex-4 User Guide UG070 (v1.4) September 12, 2005.31. Xilinx, Using Digital Clock Managers (DCMs) in Spartan-3 FPGAs XAPP462 (v1.0) July 9, 2003.

from the qubit to the quantum search algorithms

Documents